Double field theory

Abstract

The zero modes of closed strings on a torus -the torus coordinates plus dual coordinates conjugate to winding number-parameterize a doubled torus. In closed string field theory, the string field depends on all zero-modes and so can be expanded to give an infinite set of fields on the doubled torus. We use string field theory to construct a theory of massless fields on the doubled torus. Key to the consistency is a constraint on fields and gauge parameters that arises from the L 0 −L 0 = 0 condition in closed string theory. The symmetry of this double field theory includes usual and 'dual diffeomorphisms', together with a T-duality acting on fields that have explicit dependence on the torus coordinates and the dual coordinates. We find that, along with gravity, a Kalb-Ramond field and a dilaton must be added to support both usual and dual diffeomorphisms. We construct a fully consistent and gauge invariant action on the doubled torus to cubic order in the fields. We discuss the challenges involved in the construction of the full nonlinear theory. We emphasize that the doubled geometry is physical and the dual dimensions should not be viewed as an auxiliary structure or a gauge artifact.

Doubled α ′-geometry

“…But this approach can be generalized [10][11][12] In more recent developments [14] the construction of such a double field theory was based on closed string field theory [15,16]. This work identified the constraints mentioned above as the strong version of the L 0 −L 0 = 0 level-matching condition of closed string fields.…”

“…On the other hand, the string field theory based analysis [14] is by construction duality covariant (although background dependent), suggesting again that duality need not be corrected. It was noted in [23], moreover, that the gauge symmetry brackets calculated to lowest order in derivatives in [14] receive computable α ′ corrections.…”

“…It was noted in [23], moreover, that the gauge symmetry brackets calculated to lowest order in derivatives in [14] receive computable α ′ corrections. These corrections have been determined, appear to agree with the results to be presented here, and will be considered elsewhere as supporting evidence for the connection to string theory.…”

“…(For this paper we do not compactify, so these constraints eliminate winding modes.) We then use the chirality constraint to eliminate half the oscillator modes: writing the metric as 14) in terms of the usual D-valued spacetime indices m, we have chirality:…”

“…Since tr(V ′ ) = 6 div V we also have 14) which states that the trace of a trivial tensor gives no contribution to the action. Since T = T + V ′ for some V , we also have that…”